The Hadamard Maximal Determinant Problem
Papers

R. P. Brent, J. H. Osborn, W. P. Orrick and P. Zimmermann,
Maximal determinants and saturated Doptimal designs
of orders 19 and 37,
submitted (currently under revision).
Preliminary version
available from
arXiv 1112.4160v1.

R. P. Brent,
Finding Doptimal designs by randomised decomposition and
switching,
Australasian Journal of Combinatorics 55 (2013), 1530.
Also arXiv 1112.4671v5.
Erratum.

R. P. Brent and J. H. Osborn,
General lower bounds on maximal determinants of binary
matrices,
The Electronic Journal of Combinatorics 20(2), 2013, #P15, 12 pp.
Also
arXiv:1208.1805v6.

R. P. Brent and J. H. Osborn,
Bounds on minors of binary matrices,
Bulletin of the Australian Math. Soc. 88 (2013), 280285.
Also (longer version)
arXiv:1208.3330v3.

R. P. Brent and J. H. Osborn,
On minors of maximal determinant matrices,
Journal of Integer Sequences 16 (2013), article 13.4.2, 30 pp.
Also arXiv:1208.3819v3.
For relevant data files, see below.

R. P. Brent, J. H. Osborn and W. D. Smith,
Lower bounds on maximal determinants of +1 matrices via
the probabilistic method.
Available from
arXiv:1211.3248v3,
5 May 2013.
[Largely superseded by the following.]

R. P. Brent, J. H. Osborn and W. D. Smith,
Lower bounds on maximal determinants of binary matrices via the
probabilistic method.
Available from
arXiv:1402.6817v2,
26 Jan. 2015.

R. P. Brent, J. H. Osborn and W. D. Smith,
Probabilistic lower bounds on maximal determinants of binary matrices,
Australasian Journal of Combinatorics 66 (2016), 350364.
Also (slightly revised version)
arXiv:1501.06235v7,
25 Oct. 2016.
For a longer version see
arXiv:1402.6817v6.
Talks
Talks by Richard Brent
Workshop on Probabilistic and Extremal Combinatorics,
Monash University, 1317 June 2016

Probabilistic lower bounds on maximal determinants of binary matrices
(joint work with Judyanne Osborn and Warren Smith).
Overhead transparencies (beamer format):
[pdf]
Gene Golub Day Memorial Workshop, Hong Kong Baptist University,
7 February 2015

Lower bounds for the Hadamard maximal determinant problem
(joint work with Judyanne Osborn and Warren Smith).
Overhead transparencies (beamer format):
[pdf]
57th Annual Meeting of the Australian Mathematical Society,
Sydney, 1 October 2013

Improved lower bounds on the Hadamard maxdet problem, Part I
(joint work with and presented by Judyanne Osborn).
Overhead transparencies (beamer format):
[pdf]
Preprint:
arXiv:1309.2795v2

Improved lower bounds on the Hadamard maxdet problem, Part II
(joint work with Judyanne Osborn and Warren Smith).
Overhead transparencies (beamer format):
[pdf]
Preprint (March 2014):
arXiv:1402.6817v2
36th ACCMCC Conference, University of NSW, 1014 December 2012

Lower bounds on maximal determinants via the probabilistic method
(joint work with Warren Smith and Judyanne Osborn)
We use the probabilistic method to give improved lower bounds on the
maximal determinant of n times n {+1} matrices.
Overhead transparencies (beamer format):
[pdf]
56th Annual Meeting of the Australian Mathematical Society,
Ballarat, 26 September 2012

Lower bounds on maximal determinants
(joint work with Judyanne Osborn)
We give lower bounds on the maximal determinant of
n times n
{+1} matrices, both with and without the assumption of the
Hadamard conjecture.
Overhead transparencies (beamer format):
[pdf]
Hadamard Workshop in Honour of Kathy Horadam, RMIT, Melbourne,
28 November 2011

Finding many solutions of the Hadamard maximal determinant
problem given the maximal Gram matrices
(presented at the International Workshop on
Hadamard Matrices and their Applications
in Honour of Kathy Horadam's sixtieth birthday,
RMIT University, Melbourne, Australia,
2830 November, 2011).
We describe how randomised decomposition and switching
can be used to find many solutions of the Hadamard maximal
determinant problem for certain orders.
Revised overhead transparencies (beamer format):
[pdf]
Hadamard Maximal Determinant Workshop, ANU, 14 May 2010

Decomposing Gram matrices,
(invited talk presented at the
Hadamard Maximal Determinant Workshop,
Australian National University, Canberra, 1317 May 2010).
We discuss algorithms to decompose Gram matrices in
order to find {+1,1} matrices of maximal determinant.
Overhead transparencies (beamer format):
[pdf]
Talks by Will Orrick and Paul Zimmermann
 Talks by Will Orrick are available
here.
 Talks by Paul Zimmermann are available
here.
Data for Minors of Maxdet Matrices
The following links are relevant to the paper
On minors of maximal determinant matrices.
In the pdf files, frequencies (multiplicities) of minors
are indicated by superscripts. For example, 2^{336} means
that the minor with absolute value 2 occurs 336 times.
 Minors and frequencies for n = 0 mod 4,
n ≤ 16: pdf;
n = 20: txt.
 Minors and frequencies for n = 1 mod 4,
n ≤ 17: pdf.
 Minors only for
n = 21: txt.
 Minors and frequencies for n = 2 mod 4, n ≤ 18:
pdf.
 Minors and frequencies for n = 3 mod 4,
n ≤ 15: pdf;
n = 19: pdf.
Data for Various Orders
Gram matrices are encoded in a compressed format, which is
described here.
A C program to convert to compressed format is here,
and a C program to convert from compressed format is
here.
The following links are relevant to the orders 19, 26, 27, 29, 33 and 37
that are considered in the above papers/talks.
Order 19
Order 26
Order 27
Order 29
Order 33
Order 37
Other Online Resources
A great resource is
Will Orrick's page.
Return to Richard Brent's index page